Gravitational collapse of a self-gravitating unidirectional fluid flow

Highlights

• The phenomena of gravitational collapse of a unidirectional isotropic matter configuration.

• The junction conditions for a static exterior geometry and the non-static interior geometry are evaluated.

• We have calculated the time lapse for the appearance of the black hole and cosmological horizon.

• We observed that in the de-Sitter space, the unidirectional perfect fluid flow does not make the collapse disappear.

• We fund that the presence of string tension increases the time lapse of horizon formation.

Abstract

This paper explores the collapsing process of a unidirectional isotropic matter configuration. The junction conditions for a static exterior geometry and the non-static interior geometry are expressed in terms of the cosmological constant. The time-lapse for the appearance of the black hole and the cosmological horizon is calculated. It is observed that in the de-Sitter space, the unidirectional perfect fluid flow does not make the collapse disappear. The vacuum energy of the cosmological constant makes the collapsing process quite slow and affects the time-lapse of horizon formation. Moreover, the presence of string tension increases the time-lapse of horizon formation.

Introduction

General Relativity (GR) represents gravitational force by space-time curvature. This theory is based on the Einstein field equations (EFEs) relating the geometry and matter of space-time. In geometric units ($c=1=G$), these equations can be expressed alternatively as

$$R_{\mu \nu }=8\pi (T_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }T)-\Lambda g_{\mu \nu },$$

where $g_{\mu \nu },R_{\mu \nu }$and $T_{\mu \nu }$are components of the metric tensor, Ricci curvature tensor and energy-momentum tensor. The quantity $R$is the Ricci scalar whereas $\Lambda$is the well known cosmological constant (CC).

The exact solutions of EFEs include space-time singularities which may appear as outcomes of collapse of massive stars (Thorne, 1965, Shapiro, Teukolsky, 1991) which is the most significant mechanism that may result in several equilibrium states. These states include white dwarfs, neutron stars, and singularity (Penrose, 1965). These singularities may be either covered (a black hole) or naked. The singularity theorems (Hawking, 1967) and cosmic censorship conjecture (Penrose, 1969) discussed these two kinds of singularities and indicate that during the collapsing process, the trapped surfaces may form that lead to space-time singularity. Yet, these theorems fail to provide evidence for the visibility of the singularity. The Penrose cosmic censorship conjecture predicts a singularity covered by an horizon as a result of collapse (Penrose, 2002). Since there is no theorem regarding the cosmic censorship, therefore it is appealing to study the collapsing process in different scenarios to explore the results as naked singularity or a black hole.

There is a lot of literature in GR that demonstrate space-time singularities and gravitational collapse (Joshi, Dadhich, Maartens, 2002, Goswami, Joshi, Vaz, Witten, 2004). The Incoherent dust in the interior of a star is the most convenient model. The collapse of incoherent dust discussed in different space-times, yields different results (Joshi, Dwivedi, 1999, Nakao, Iguchi, Harada, 2001, Ghosh, Deshkar, 2007). Indeed, the star’s interior is filled with fluid (dust with pressure) and explains the collapse in a more realistic way. This initiated the study of collapsing processes with perfect fluid filled in the interior of stellar systems (da Rocha, Wang, Santos, 1999, Herrera, Santos, 2005, Goswami, Joshi, 2004). The perfect fluid is an ideal fluid which is thought-provoking to find the outcomes of a gravitational collapse with a realistic matter like anisotropic fluids which are either cold or may contain some heat contents (Pinheiro, Chan, 2008, Di Prisco, Herrera, MacCallum, Santos, 2009).

The study of the collapsing process with spherical symmetric space-time involving CC is another way towards the physical scenarios. The CC represents the energy density in vacuum space. It was introduced to hold back gravity and attain a fixed universe Nussbaumer (2014). Thus, it acts as an anti-gravity term which controls the growth of the universe and problems regarding structure and age of massive bodies. Markovic and Shapiro (2000) studied the gravitational dust collapse with Schwarzschild geometry as exterior region and Friedman metric as interior region, in presence of positive valued CC. Lake (2000) did the same work with negative value of CC to explore its influence on the process of gravitational collapse. Cissoko et al. (1998), studied the influence of CC on the dust collapse with static and non-static background metrics. Ahmad and Malik (2016) investigated the influence of CC on collapsing phenomena for anisotropic matter configuration with spherically symmetric space-time. The consequences of the positive valued CC on a perfect fluid collapse in spherical symmetric space-time has been discussed by Sharif and Ahmad (2007). They found that CC makes the collapsing phenomenon quiet and binds the area of black hole.

Yousaf and his collaborators (Yousaf, Bamba, Bhatti, 2016, Yousaf, Bhatti, Farwa, 2017, Bhatti, Yousaf, 2017) deduced the inhomogeneity factors during celestial collapse and claimed that extra curvature ingredients of extended gravity theories are responsible for the system to leave or enter the uniform phase. Herrera et al. (1998) claimed crucial effects of imperfect fluid on the energy-density inhomogeneity during the evolution of spherical stars and found that an inhomogeneous situation leads to the collapsing stage. Herrera (2011) explored inhomogeneity factors for non-adiabatic as well as adiabatic collapsing structure and found that for the a stable configuration of a system must satisfy certain conditions. A lot of attention has been made in the systematic investigation of collapsing processes in non-tilted, tilted, as well as conformally flat (Bhatti, 2018, Bhatti, Bamba, Nawaz, Yousaf, 2019, Bhatti, Tariq, 2019, Bhatti, Yousaf, Zarnoor, 2019, Bhatti, Tariq, 2020, Bhatti, Yousaf, Khadim, 2020) relativistic systems with different fields. Yousaf and his collaborators (Yousaf, Bhatti, 2016, Yousaf, Bhatti, Ilyas, 2018, Yousaf, Bhatti, Malik, 2019a, Yousaf, Bhatti, Saleem, 2019b, Yousaf, Bhatti, Yaseen, 2019c, Yousaf, 2020) found that extra curvature effects other than the usual GR could be referred as a vital framework to describe emerging solutions for collapsing models.

In this paper, our aim is to explore the influence of CC in the analysis of gravitational collapse with a perfect fluid attached with strings (Letelier, 1980, Stachel, 1980) as an imperfect fluid. The stress energy tensor for such distribution is given by

$$T_{\nu \mu }=(p+\rho )u_{\nu }{u}_{\mu }-p{g}_{\nu \mu }-\lambda x_{\nu }{x}_{\mu },$$

where the energy density for the above mentioned matter configuration is indicated via $\rho$with attached one dimensional particles (strings) while $p$represents the pressure and $\lambda$is the string tension density. The equation of motion (1)coupled with (3) could describe Einstein equations for the relativistic anisotropic matter having pressure different from zero in the direction of $x^{\alpha }$under the constraint $\lambda <0$. The fluid admits a unidirectional flow along the attached radially directed strings, i.e., $x$-direction. The velocity and position vector components, $u_i$and $x_i,$satisfy conditions

$$u^{\mu }{u}_{\mu }=-x^{\mu }{x}_{\mu }=1,u^{\mu }{x}_{\mu }=0.$$

As we have assumed the comoving coordinate system, therefore the form of $x^{\mu }$can be written explicitly as follows

$$x^{\mu }=X^{-1}{\delta }_1^{\alpha }.$$

This fluid possesses the following properties:

• ${\rho }_p$is the particle density of the configuration, $\rho ={\rho }_p+\lambda .$If $\lambda$is absent, only particles contribute in the baryonic energy density of the fluid.

• This fluid (Amirhashchi and Zainuddin, 2010a) is called stiff for $\rho -p=0,$and anti-stiff for $p+\rho =0.$

• The strings (Amirhashchi and Zainuddin, 2010b) may be considered to be Reddy ($\rho +\lambda =0$) or Nambu (Geometric) ($\rho -\lambda =0$).

As a more realistic approach, it is worthwhile to discuss the collapse of a unidirectional perfect fluid flow inside the star in the presence of CC. The next section contains the junction conditions and discussion of space-time as well as fluid filled inside the star. Section 3 contains EFEs and their solutions. In Section 4, we explore the time lapse for the appearance of apparent horizons. The last section contains results and discussion.